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Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks

arXiv 2508.12596 ↗First posted 18 Aug 2025cs.LGcited by 0

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Abstract (arXiv)

Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework supports inputs and outputs given as a tuple of Cartesian tensors of different rank as well as spherical tensors of different type. We introduce tensor network generators for invariant maps and obtain equivariant maps via differentiation. Specifically, we derive general continuous equivariant maps from vector inputs to Cartesian or spherical tensor output. Finally, we clarify how common equivariant primitives in geometric graph neural networks arise within our construction.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 20 of its 40 references that are in the map. First posted on arXiv 18 Aug 2025.

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20 of 40 references.

References · 20

  1. 2021E(n) Equivariant Graph Neural Networkscited by 965
  2. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  3. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 441
  4. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  5. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 305
  6. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 264
  7. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 228
  8. 2022e3nn: Euclidean Neural Networkscited by 174
  9. 2018Universal approximations of invariant maps by neural networkscited by 127
  10. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  11. 2023Tensor Networks Meet Neural Networks: A Survey and Future Perspectivescited by 32
  12. 2019Einconv: Exploring Unexplored Tensor Network Decompositions for Convolutional Neural Networkscited by 25
8 more
  1. 2017Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Designcited by 24
  2. 2022Equivariant Graph Attention Networks for Molecular Property Predictioncited by 20
  3. 2022Brauer's Group Equivariant Neural Networkscited by 17
  4. 2022Machine learning and invariant theorycited by 13
  5. 2024Tensor cumulants for statistical inference on invariant distributionscited by 8
  6. 2022Cost-efficient Gaussian Tensor Network Embeddings for Tensor-structured Inputscited by 5
  7. 2022Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalismcited by 3
  8. 2024Tensor learning with orthogonal, Lorentz, and symplectic symmetriescited by 3

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